physics pp. 501–512
Theoretical study of two-element array of
equilateral triangular patch microstrip antenna on ferrite substrate
K K VERMA and K R SONI
Microwave Laboratory, Department of Physics, Malaviya National Institute of Technology, Jaipur 302 017, India
E-mail: kkvermaphd@rediffmail.com
MS received 6 November 2004; revised 11 April 2005; accepted 28 April 2005
Abstract. The radiation characteristics of a two-element array of equilateral triangular patch microstrip antenna on a ferrite substrate are studied theoretically by considering the presence of bias magnetic field in the direction of propagation of electromagnetic waves. It is found that the natural modes of propagation in the direction of magnetic field are left- and right-circularly polarized waves and these modes have different propagation constants. In loss-less isotropic warm plasma, this array antenna geometry excites both electromagnetic (EM) and electroacoustic plasma (P) waves in addition to a nonradiating surface wave. In the absence of an external magnetic field, the EM- and P-waves can be decoupled into two independent modes, the electroacoustic mode is longitudinal while the electromagnetic mode is transverse. The far-zone EM-mode and P-mode radiation fields are derived using vector wave function techniques and pattern multiplication approaches.
The results are obtained in both plasma medium and free space. Some important antenna parameters such as radiation conductance, directivity and quality factor are plotted for different values of plasma-to-source frequency.
Keywords. Microstrip array antenna; ferrite substrate; radiation properties; plasma.
PACS Nos 84.40.Ba; 52.40.Fd
1. Introduction
In recent years, ferrite substrates have been the subject of much interest for mi- crostrip patch antennas and their arrays. The high dielectric constant of the ferrite substrates reduces the antenna dimensions and when biased with a DC magnetic field, the antenna exhibits a number of novel properties. These include frequency tuning agility, the generation of circular polarization, reduction of surface waves and radar cross-section control. Ferrite materials are also used to generate beam scanning antennas. Microstrip antennas mounted on aerospace vehicles encounter plasma medium during their travel in space, as a result of which radiation prop- erties are altered significantly. This change is caused due to the generation of electroacoustic (P) waves in addition to electromagnetic (EM) waves [1–5].
Table 1. Design requirement and substrate characteristics of the ferrite sub- strate Ni1.062Co0.02Fe1.948O4.
Design frequency (f) 1.0 GHz
Relative permittivity (εr) 14.78
Dieletric loss tangent (tanδe) 0.0005 Magnetic loss tangent (tanδm) 0.005
Applied DC magnetic bias field (H0) 7.96×104 A/m Saturation magnetization (µ0Ms) 0.03 T
Patch dimension (a) 0.022 m
Substrate thickness (h) 0.0016 m
Gyromagnetic ratio (γ) 1.76×1011rad /s·T
Ferrite and other magnetic materials have been extensively used in several mi- crowave devices such as phase shifters, isolators, circulators, tunable filters, delay lines etc. Ferrite materials have a significant amount of anisotropy at microwave frequencies. This anisotropy gets induced by applying external DC magnetic field in ferrite or gyrotropic materials and brings about non-reciprocal behavior in them.
This paper describes the radiation characteristics of a two-element array of equilateral triangular patch microstrip antenna on the ferrite substrate Ni1.062Co0.02Fe1.948O4. Design requirement and substrate characteristics consid- ered for this analysis are listed in table 1.
2. Radiation field expressions
The top and side view of the configuration of array antenna is shown in figure 1. It consists of two identical triangular microstrip patch elements of arm lengtha, on a ferrite substrate Ni1.062Co0.02Fe1.948O4 of thicknesshand substrate permittivity εr. The array elements are separated by a distanced and the progressive phase excitation difference between the patches is β1. Each patch can be excited by a microstrip transmission line connected to the edge or by a coaxial line from the back at the plane φ = 0. We have considered the patch as a cavity which acts as a disc resonator. Among the various modes that may be excited in such a disc resonator, we have considered TMmn mode with respect toz-axis. Herenand m are the mode numbers associated withxandy directions respectively.
TheEz component of the field inside the cavity for dominant mode is given as Ez=A1,0,−1[2 cos(2πx/√
3a+ 2π/3) cos(2πy/3a) + cos(4πy/3a)]. (1) When the propagation of electromagnetic waves takes place along the direction of applied magnetic bias field, two plane wave modes – the left-hand circularly polarized (LHCP) mode and the right-hand circularly polarized (RHCP) mode may exist. The magnetic properties of the ferrite substrate affect both these modes.
Therefore, the effective permeability for right-hand circular polarization and left- hand circular polarization will be given as [6–8]
µeffR=µ01 +ωm
ω0−ω, (2)
Figure 1. (a) 2D top view of the configuration of two-element triangular patch array microstrip antenna on ferrite substrate. (b) Side view of the configuration of two-element triangular patch microstrip array antenna on ferrite substrate. (c) 3D view of the configuration of two-element triangular patch microstrip array antenna on ferrite substrate.
µeffL=µ01 +ωm
ω0+ω. (3)
The expression of the resonant frequency of the considered equilateral triangular microstrip antenna on ferrite substrate in TMmnmode is given as
fr= 2c√ µ0
3a√ εrµeff
. (4)
Other field components are obtained by solving Maxwell’s equations. By image theory, the ground plane may be replaced by an image of the top conductor. The magnetic currents also exist along the edges of triangular conductor and may be evaluated from
M= 2(E×n), (5)
wherenis a unit vector normal to the aperture.
Using linearized hydrodynamic theory of plasma, vector wave function technique and neglecting the coupling between the elements, the basic equations of far-zone fields of the two-element array of triangular microstrip patch antenna are given as [9–11]
Eθt=−ιη0ω[−Fxsinφ+Fycosφ]
·exp{−ι(βer1+ 0.5β1)}
r1
+exp{−ι(βer2−0.5β1)}
r2
¸
, (6)
Eφt=ιη0ω[Fxcosθcosφ+Fycosθsinφ]
·exp{−ι(βer1+ 0.5β1)}
r1
+exp{−ι(βer2−0.5β1)}
r2
¸
. (7)
In the far-zone field region, the values of the radius vectorsr1,r2can be calculated from figure 1.
r1≈r+d/2 cosθ
r2≈r−d/2 cosθ, for phase variation and
r1≈r2≈r, for amplitude variation.
Applying these approximations in eqs (6) and (7), we get:
For EM-mode:
Eθt=−ιη0ω[−Fxsinφ+Fycosφ]2 cos[0.5(βedcosθ+β1)]e−iβer
r (8)
Eφt=ιη0ω[Fxcosθcosφ+Fycosθsinφ]2 cos[0.5(βedcosθ+β1)e−iβer r
(9) and for plasma-mode:
Ept= 2hβpω2p/3aωε0(ω2−ωp2) exp(−ιβpr)/r
×exp(−ιβpasinθcosφ/√
3)×2 cos[0.5(βpdcosθ+β1)×[Epx+Epy], (10)
whereEθt,Eφtare the components of total electric field vectors for EM-mode,Ept
is the total electric field vector for plasma-mode,Fx is thex-component of vector electric potential, Fy is the y-component of vector electric potential, Epx is the x-component of the electric field vector for plasma-mode,Epy is the y-component of electric field vector for P-mode, βe is the phase propagation constant for EM- mode given by 2πA/λ0, βp is the phase propagation constant for plasma-mode given byβec/v,cis the velocity of light,vis the root mean square thermal velocity of electron, A is the plasma frequency parameter given by (1−ωp2/ω02)1/2, ωp is the angular plasma frequency, ω is the angular source frequency and β1 is the progressive phase excitation difference between the patches.
3. Field patterns
The expression for total field patternR(θ, φ) is obtained as
R(θ, φ) =|Eθt|2+|Eφt|2. (11) Then, the radiation field patterns in theE-plane (φ= 0) and H-plane (φ=π/2) are given as
Re(θ, φ) =|Eθt|2+|Eφt|2=η02ω02(|Fy|2+|Fx|2cos2θ) (E-plane), (12) Rh(θ, φ) =|Eθt|2+|Eφt|2=η20ω02(|Fx|2+|Fy|2cos2θ) (H-plane).
(13) The values ofRe andRh are calculated for a case takingf = 1 GHz,a= 2.2 cm, εr = 14.78, applied DC magnetic bias field (H0) = 7.96×104 A/m, saturation magnetization (µ0Ms) = 0.03 T, d=λ/2 and the phase difference β1=π/2. The results are plotted in figures 2 and 3 for two different planes (φ= 0 andφ=π/2) forA= 1.0, i.e. in free space and in figures 4 and 5 for two different planes (φ= 0 and φ = π/2) for A = 0.5, i.e. in plasma medium. The variation of effective permeability for both RHCP and LHCP modes with applied magnetic bias field (H0) is shown in figure 6. The P-mode fields are plotted in figure 7 forA= 0.5 for a limited range of 10◦ (from 60◦to 70◦). The field patterns are also compared with single-element triangular patch microstrip antenna.
4. Other antenna parameters
4.1Radiation conductance
The total power radiated can be calculated by employing Poynting’s theorem and is given as
Pr= µ1
2
¶ µ1 2
¶ ½ Re
Z Z
S
(E×H∗)ds
¾ .
Figure 2. Variation of R(θ, φ) for A = 1.0 (free space) for single-element and two-element (β1 =π/2) equilateral triangular patch microstrip antenna on ferrite substrate inE-plane.
Figure 3. Variation of R(θ, φ) for A = 1.0 (free space) for single-element and two-element (β1 =π/2) equilateral triangular patch microstrip antenna on ferrite substrate in H-plane.
The factor 12 is due to the fact that the power is radiated through the upper half space only andS is the total spherical surface area.
Pr= µ1
4
¶ ½ Re
Z Z
S
(EθHφ∗−EφHθ∗)ds
¾ .
Thus, the expressions for radiated power in EM-modes is obtained using the rela- tion [6]:
Pe= (A/4η0) Z 2π
0
Z π
0
{|Eθt|2+|Eφt|2}r2sinθdθdφ. (14) Here,A= (1−ωp2/ω02)1/2andωp/ω0 is the plasma-to-source frequency ratio.
Figure 4. Variation ofR(θ, φ) forA = 0.5 (plasma) for single-element and two-element (β1 = π/2) equilateral triangular patch microstrip antenna on ferrite substrate inE-plane.
Figure 5. Variation ofR(θ, φ) forA = 0.5 (plasma) for single-element and two-element (β1 = π/2) equilateral triangular patch microstrip antenna on ferrite substrate inH-plane.
The radiation conductance of an antenna in EM-mode can be defined as
Ge= 2Pe/V02. (15)
4.2Directivity
The directivity of an antenna is defined as the ratio of the maxium radiation in- tensity (power per unit solid angle) to the average radiation intensity. It can be expressed as
De= 4π{max(|Eθt|2+|Eφ|2)}
Á ½Z 2π
0
Z π
0
|Eθt|2+|Eφt|2r2sinθdθdφ
¾
. (16)
Figure 6. Variation of effective permeability of substrate with applied bias field.
Figure 7. Variation of plasma mode field pattern|Ept|2forA= 0.5 (plasma) for single-element and two-element (β1 = π/2) equilateral triangular patch microstrip antenna on ferrite substrate.
4.3Quality factor
A parameter specifying the frequency selectivity of a resonant circuit is the quality factor Q, which can be defined as the ratio between energy stored in the system and the energy lost. The total Q of a microstrip radiating element comprises contributions due to the radiation Qr, conductor loss Qc, and dielectric loss Qd
quality factors. So
1/Qt= 1/Qr+ 1/Qc+ 1/Qd, (17)
where Qr = ω0Wt/Pr, Qc = ω0Wt/Pc = (π f µ σ)1/2h and Qd = ω0Wt/Pd = 1/tanδ. Here,Wtis the energy stored in the antenna element,Pc andPdare power
Figure 8. Variation of radiation conductance Ge for single-element and two-element equilateral triangular patch microstrip antenna on ferrite sub- strate for EM-mode with plasma parameterA.
loss factors due to the conductors and dielectric, respectively,σis the conductivity of the conductors.
The energy stored in the triangular radiating element is given by Wt= (εh/2)
Z Z
|Ez(x, y)|2dxdy. (18)
5. Conclusion
The radiation characteristics of a two-element linear array of equilateral triangular patch microstrip antenna on ferrite substrate have been studied by considering the presence of bias magnetic field in the direction of propagation of EM waves. The re- sults of the array geometry are compared with those of single-element of equilateral triangular patch microstrip antenna. It is found that there is a significant change in the radiation characteristics of array geometry. In the case of EM-mode, the shape of the field pattern has been modified to a great extent and redistributes the field intensities considerably. It is also observed that the radiation patterns of a single- element antenna contain only one major lobe of considerably wide beam-width, while the array geometry produces a directive beam with a narrow beam-width. In the case of P-mode, the field patterns are similar to single-element microstrip an- tenna. It is further observed that the values of radiation conductance are consider- ably higher for array geometry and it increases continuously with increasing plasma parameter. The values of directivity for array geometry are also considerably higher than that of the single-element of equilateral triangular patch microstrip antenna.
The values of quality factor for array geometry are lower than that of single-element of equilateral triangular patch microstrip antenna. The computed values of pattern characteristics of the present array are given in table 2. It is clear from this table that half-power beam-width of the patterns in free space is relatively small in com- parison to plasma medium. Thus it reveals that the patterns are more directive in free-space than in plasma medium. Variation of radiation conductance Ge for single-element and two-element equilateral triangular patch microstrip antenna on ferrite substrate with plasma parameter A is shown in figure 8. From this figure, it is observed that the radiation conduction (Ge) of the two-element array an- tenna is more than that of the single-element antenna for all the plasma frequency.
Table 2. Pattern characteristics of the two-element array of triangular mi- crostrip patch antenna on ferrite substrate.
φ= 0 plane φ= 0 plane φ=π/2 plane φ=π/2 plane Pattern
characteristics
A= 1.0 A= 0.5 A= 1.0 A= 0.5
RHCP LHCP RHCP LHCP RHCP LHCP RHCP LHCP
Half-power beam
width (HPBW) 60◦ 80◦ 140◦ 150◦ 60◦ 80◦ 140◦ 150◦ Full null beam
width (FNBW) 120◦ 140◦ 360◦ 280◦ 120◦ 140◦ 360◦ 280◦ Direction of max.
radiation 0◦ 0◦ 90◦ 90◦ 0◦ 0◦ 90◦ 90◦
Figure 9. Variation of directivity De for single-element and two-element equilateral triangular patch microstrip antenna on ferrite substrate for EM–
mode with plasma parameterA.
Figure 10. Variation of quality factor Qe for single-element and two-ele- ment equilateral triangular patch microstrip antenna on ferrite substrate for EM-mode with plasma parameterA.
It is maximum in free space and decreases on increasing plasma frequency. It is also observed that the radiation conduction (Ge) with LHCP waves is higher than that with RHCP waves. Variation of radiation conductance De for single-
Table 3. Antenna parameters for single-element and two-element equilateral triangular patch microstrip antenna on ferrite substrate with RHCP waves.
Gefor two- De for Defor
Plasma Ge for single element single two-element Qe for Qefor parameter element linear array element linear array single two-element
(A) (mho) (mho) (dB) (dB) element linear array
1.0 1.588×10−5 3.175×10−5 4.751 4.669 444.418 371.717 0.9 1.428×10−5 2.855×10−5 4.755 3.048 453.355 384.392 0.8 1.268×10−5 2.536×10−5 4.758 1.927 462.635 397.928 0.7 1.109×10−5 2.218×10−5 4.761 1.773 472.281 412.419 0.6 9.468×10−6 1.900×10−5 4.764 1.77 482.319 427.974 0.5 7.911×10−6 1.582×10−5 4.766 1.768 492.774 444.72 0.4 6.326×10−6 1.265×10−5 4.768 1.766 503.678 462.803 0.3 4.743×10−6 9.486×10−6 4.769 1.764 515.061 482.396 0.2 3.161×10−6 6.323×10−6 4.77 1.762 526.961 503.703 0.1 1.580×10−6 3.161×10−6 4.771 1.98 539.417 526.965
0.0 0.0 0.0 4.771 4.771 – –
Table 4. Antenna parameters for single element and two-element equilateral triangular patch microstrip antenna on ferrite substrate with LHCP waves.
Ge for two- Defor De for
Plasma Ge for single element single two-element Qefor Qefor parameter element linear array element linear array single two-element
(A) (mho) (mho) (dB) (dB) element linear array
1.0 2.811×10−5 5.621×10−5 4.736 7.395 449.784 355.889 0.9 2.526×10−5 5.052×10−5 4.742 6.667 462.144 371.617 0.8 2.242×10−5 4.484×10−5 4.748 5.45 475.146 388.724 0.7 1.959×10−5 3.919×10−5 4.754 3.585 488.847 407.407 0.6 1.678×10−5 3.356×10−5 4.758 1.896 503.311 427.904 0.5 1.397×10−5 2.794×10−5 4.762 1.772 518.612 450.503 0.4 1.117×10−5 2.233×10−5 4.765 1.769 534.831 475.559 0.3 8.370×10−6 1.674×10−5 4.768 1.766 552.062 503.507 0.2 5.578×10−6 1.116×10−5 4.77 1.763 570.413 534.897 0.1 2.788×10−6 5.576×10−6 4.771 2.688 590.006 570.422
0.0 0.0 0.0 4.771 4.77 – –
element and two-element equilateral triangular patch microstrip antenna on ferrite substrate with plasma parameter A is shown in figure 9. From this figure, it is observed that the directivity (De) of this array with LHCP waves is higher than that with RHCP waves for free space and all plasma frequency. Variation of radiation conductance Qe for single-element and two-element equilateral triangular patch microstrip antenna on ferrite substrate with plasma parameterAis shown in figure 10. From this figure, it is noticed that the low values of quality factor (Qe) of this array antenna in free space indicate that antenna is radiating power more effectively
in free space. Hence at low values ofA, more energy is radiated in the form of plasma waves, which increases the quality factor of antenna in plasma medium. If we are considering the effect of LHCP and RHCP modes, then the total quality factor of this array antenna with LHCP waves is lower than that with RHCP waves.
Finally, it is concluded that two-element linear array of equilateral triangular patch microstrip antenna has unique radiation characteristics and can be employed in applications where high gain and narrow beam-width are required. The results of the present study are useful, particularly for space vehicles because such type of array antenna can be mounted on the flat surface as well as on the curved surface of the vehicles.
References
[1] I L Freeston and R K Gupta,Proc. IEE 18, 633 (1971)
[2] J R James and G J Wilson,Microwaves optics and acoustic 1, 165 (1977)
[3] J Helszajn and D S James,IEEE Trans. Microwave Theory and TechniquesMTT-26, 95 (1978)
[4] Y T Lo, D Soloman and W F Richards,IEEE Trans. Antenna and Propagation AP- 27, 137 (1979)
[5] K R Carver and J W Mink,IEEE Trans. Antenna and PropagationAP-29, 2 (1981) [6] I J Bahl and P Bhartia,Microstrip antennas (Artech House, London, 1980)
[7] C A Balanis, Antenna Theory – analysis and design (Harper and Row, New York, 1982)
[8] R E Collin,Foundations for microwave engineering (McGraw Hill, New York, 1992) [9] K K Verma and K R Soni,Indian J. Phys.B78, 1359 (2004)
[10] K K Verma and K R Soni,Indian J. Phys.B78, 1397 (2004) [11] K K Verma and K R Soni,Pramana – J. Phys.64, 147 (2005)
